I am not sure if I am stupid, or the problem is missing something...

[pineapplewithmouse]

"The diagram shows a graph of the function f(x) (In the picture I sent)
Draw a graph of the function g(x) (not necessarily a linear function)
so that the equation f(x) = g(x):
A. Will have one solution
B. Will have an infinity of solutions
C. Will have no solutions"

If g(x) is equal to f(x) it means that the function is the same, so there will be an infinity of solutions. In order for there to be one solution, the function must be a constant function (and it is not) And so that will have no solution the slope should be undefined (vertical, and is also not) How do I solve this question? Please help me...

 

[Deleted member 4993]

"The diagram shows a graph of the function f(x) (In the picture I sent)
Draw a graph of the function g(x) (not necessarily a linear function)
so that the equation f(x) = g(x):
A. Will have one solution
B. Will have an infinity of solutions
C. Will have no solutions"

If g(x) is equal to f(x) it means that the function is the same, so there will be an infinity of solutions. In order for there to be one solution, the function must be a constant function (and it is not) And so that will have no solution the slope should be undefined (vertical, and is also not) How do I solve this question? Please help me...

View attachment 27918

I agree with your following conclusion

If g(x) is equal to f(x) it means that the function is the same, so there will be an infinity of solutions.

 

[lev888]

"The diagram shows a graph of the function f(x) (In the picture I sent)
Draw a graph of the function g(x) (not necessarily a linear function)
so that the equation f(x) = g(x):
A. Will have one solution
B. Will have an infinity of solutions
C. Will have no solutions"

If g(x) is equal to f(x) it means that the function is the same, so there will be an infinity of solutions. In order for there to be one solution, the function must be a constant function (and it is not) And so that will have no solution the slope should be undefined (vertical, and is also not) How do I solve this question? Please help me...

View attachment 27918

There is some confusion going on here. f(x) = g(x) is an equation. It's not a statement that "g(x) is equal to f(x)".
We equate the expressions for f(x) and g(x) making an equation. Solving it will give us the points where the graphs intersect. You need to come up with g(x), such that A. the graphs intersect in one point, etc.

 

[Deleted member 4993]

There is some confusion going on here. f(x) = g(x) is an equation. It's not a statement that "g(x) is equal to f(x)".
We equate the expressions for f(x) and g(x) making an equation. Solving it will give us the points where the graphs intersect. You need to come up with g(x), such that A. the graphs intersect in one point, etc.

I would interpret f(x) = g(x) as a statement describing two congruent (identical) curves (functions) which are "equal" for every 'x'.

Then again I am an engineer - prone to simplistic interpretation (often 'mathematically incorrect") of mathematical statements.

 

[lev888]

I would interpret f(x) = g(x) as a statement describing two congruent (identical) curves (functions) which are "equal" for every 'x'.

Then again I am an engineer - prone to simplistic interpretation (often 'mathematically incorrect") of mathematical statements.

The problem says "the equation f(x) = g(x)".

 

[lex]

Yes, I agree if we are talking about solutions, f(x) = g(x) is not intended to be an identity.
(Of course, identity of f and g is sufficient but not necessary for an infinity of 'solutions').

 

[topsquark]

Okay, so if g(x) is not necessarily the same as f(x) for all values of x, can you think of a function g(x) such that it cuts the graph of f(x) only once?

The fun one if going to be the second question: Can you find a g(x) that cuts f(x) an infinite number of times, without g(x) = f(x) for all x.

-Dan

 

[Steven G]

To find the points of intersections between two functions, f(x) and g(x), we set them equal to one another. This does not mean that f(x) = g(x) is an identity (same graph), it can be a conditional statement (f(x)=g(x) only for some x-values but not for other x-values)

 

[Harry_the_cat]

Okay, so if g(x) is not necessarily the same as f(x) for all values of x, can you think of a function g(x) such that it cuts the graph of f(x) only once?

The fun one if going to be the second question: Can you find a g(x) that cuts f(x) an infinite number of times, without g(x) = f(x) for all x.

-Dan

The question asks to DRAW the function g(x) for each of those 3 conditions. The second condition could be met by a sinusoidal graph with y=f(x) as the equilibrium line (with a small enough amplitude to make sure it is still a function). Easy to draw, but more difficult to find the equation (which isn't asked for).

 

[JeffM]

Students are likely to be confused when helpers disagree. And I must admit that I am not 100% sure that I know what the proper usage of mathematical notation is here. But my impression is that in ACTUAL USE, mathematicians do say

[MATH]f(x) = g(x)[/MATH] when they mean [MATH]f(x) \equiv g(x).[/MATH]
It’s usually an immaterial point about notation because

[MATH]f(x) \equiv g(x) \iff f(x) = g(x) \text { for all x}[/MATH],

but it is not immaterial WHEN IT CONFUSES STUDENTS.

 

[Steven G]

Jeff, I disagree with you here. If I ask you to find the points of intersection for the functions f(x) and g(x) the 1st thing you would do is solve the equation f(x) = g(x).

If I write that 3x+5 = 8 does that mean that 3x+5 always equal 8. Of course not, it is a conditional equation. It depends on what values x is. If x=5, the the equation is not valid. However when x=1, the the equation is valid.

In my opinion, the problem never said that f(x) = g(x) but rather it said find (ok draw) g(x) such that
the equation f(x) = g(x) (a) will have one solution, (b) will have an infinity of solutions and (c) will have no solutions.

 

[Steven G]

I would interpret f(x) = g(x) as a statement describing two congruent (identical) curves (functions) which are "equal" for every 'x'.

Then again I am an engineer - prone to simplistic interpretation (often 'mathematically incorrect") of mathematical statements.

Subhotosh, the first thing one writes when they are finding the points of intersections for the two curves f(x) and g(x) is f(x)=g(x). If in fact the functions are not identical should the student then write that the equation is not valid. Then how do they find the point of intersection?

As you know by now, I hate to see equal signs that are not valid. In this case I have no problem because no one is saying (or at least should not be saying) that the equation f(x)=g(x) is an identity. Engineer or not, why would you think this?

 

[Steven G]

I am rethinking my opinion on this. When one wants the point of intersection of two functions, then they should say set f(x) = g(x).
I do not think that it is a big issue to just say f(x)=g(x).

 

[Deleted member 4993]

Subhotosh, the first thing one writes when they are finding the points of intersections for the two curves f(x) and g(x) is f(x)=g(x). If in fact the functions are not identical should the student then write that the equation is not valid. Then how do they find the point of intersection?

As you know by now, I hate to see equal signs that are not valid. In this case I have no problem because no one is saying (or at least should not be saying) that the equation f(x)=g(x) is an identity. Engineer or not, why would you think this?

Being a scientist/Engineer, I would precisely state:

At the point/s of discrete intersection, f(x) = g(x)

These distinctions are very important in cases of discrete functions. In engineering, we have to deal with discrete functions quite often (FFT).